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D e z e m b r o 2 0 1 2
410
CHOOSING AN OPTIMAL INVESTMENT STRATEGY: THE ROLE
OF ROBUST PAIR-COPULAS BASED
PORTFOLIOS
Beatriz Vaz de Melo Mendes Daniel S. Marques
Relatórios COPPEAD é uma publicação do Instituto COPPEAD de Administração da Universidade Federal do Rio de Janeiro (UFRJ)
Editor
Leticia Casotti Editoração Lucilia Silva Ficha Catalográfica Marisa Rodrigues Revert
Mendes, Beatriz Vaz de Melo. Choosing an optimal investment strategy: The role of robust
pair-copulas based portfolios / Beatriz Vaz de Melo Mendes, Daniel S. Marques. – Rio de Janeiro: UFRJ /COPPEAD, 2012.
23 p.; 27cm. – (Relatórios COPPEAD; 410)
ISBN 978-85-7508-097-9
ISSN 1518-3335
1. Finanças. 2. Cópulas (Estatística matemática). I. Marques, Daniel S. II. Título. IV. Série.
CDD – 332
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Choosing an optimal investment strategy:
The role of robust pair-copulas based portfolios
Beatriz Vaz de Melo Mendesab1, Daniel S. Marquesa
a COPPEAD, Federal University at Rio de Janeiro, phone: 55 21 25989898, P.O. Box 68514, Rio
de Janeiro, RJ, 21941-972, Brazil.
b IM, Federal University at Rio de Janeiro. Av. Athos da Silveira Ramos 149, CT, P.O. Box
68530 21941, Rio de Janeiro, RJ, 21941-909, Brazil.
Abstract
This paper is concerned with the efficient allocation of a set of financial assets and its successful
management. Efficient diversification of investments is achieved by inputing robust pair-copulas
based estimates of the expected return and covariances in the mean-variance analysis of Markowitz.
Although the whole point of diversifying a portfolio is to avoid rebalancing, very often one needs to
rebalance to restore the portfolio to its original balance or target. But when and why to rebalance
is a critical issue, and this paper investigates several managers’ strategies to keep the allocations
optimal. Findings for an emerging market target return and minimum risk investments are highly
significant and convincing. Although the best strategy depends on the investor risk profile, it
is empirically shown that the proposed robust portfolios always outperform the classical versions
based on the sample estimates, yielding higher gains in the long run and requiring a smaller number
of updates. We found that the pair-copulas based robust minimum risk portfolio monitored by a
manager which checks its composition twice a year provides the best long run investment.
Keywords: Pair-copulas; Optimal financial portfolios; Robust estimation; Rebalancing.
JEL Classifications: C01, C19,C32, C51, C58, G11.
1Corresponding author (Permanent address): Beatriz Vaz de Melo Mendes, phone: 55 21 22652914,
Rua Marquesa de Santos, 22, apt. 1204, Rio de Janeiro, 22221-080, RJ, Brazil. Email: [email protected].
1
1 Introduction
Financial institutions and portfolio managers are primarily concerned with the efficient
allocation and monitoring of sets of financial assets. Periodic portfolio rebalancing, aiming
to restore the investment back to its desired target risk and return, is a crucial step in the
process of controlling risk. Commonly asked questions are how often a portfolio should
be rebalanced, and which would be the best indicators of changes in the global economy
or in the balance among the component assets.
Efficient diversification of investments based on the mean-variance analysis of Markowitz
(1952) is widely used by institutional investors. Statistically, the resulting efficient frontier
just relies on the estimates of the expected return and covariance matrix, and the sample
estimates are the usual inputs. However, the statistical good properties of the sample
estimates are attached to the highly improbable assumption of multivariate normality.
A better characterization of the data underlying multivariate distribution will provide
more reliable estimation of the efficient frontier. This means we must know not just the
marginal univariate series behavior and their correlations, but their whole d-dimensional
probability distribution. This may be accomplished by modeling the data through pair-
copulas (Frigessi and Bakken (2007); Min and Czado (2008); Berg and Aas (2008); Fischer
et al. (2008)).
A pair-copula construction is just a hierarchical decomposition of a multivariate copula
into a cascade of bivariate copulas. Since an appropriate copula function can be found
for any type of association — linear, nonlinear, ranging from perfect negative to perfect
positive dependence — one can expect the model to truthfully represent the data at hand.
Estimation takes place at the level of the two-dimensional data, therefore avoiding the
famous curse of dimensionality.
The analysis of financial data from emerging markets poses some specific challenges.
Atypical points in transaction prices (from non-confirmed unexpected news, market ma-
nipulation, and so on) distort classical statistical inference, corrupting the inputs to the
mean-variance algorithm. A distorted correlation matrix and inflated risk estimates will
provide misleading allocations. To handle deviations from the true underlying distribution,
robust methodologies are called for. We suggest to apply the robust estimates for pair-
copulas models, initially proposed in Mendes et al. (2007). For each parametric copula
family there exist a robust weighted minimum distance or a weighted maximum likelihood
estimator providing accurate estimates under contaminations. The robust portfolios are
obtained by inserting the robust pair-copulas based mean and covariance estimates in the
mean-variance Markowitz procedure.
2
Robust methods typically detect the pattern implied by the vast majority of the data,
providing more stable estimates. Robust allocations are resistant to unjustified sudden
fluctuations of the market, which are identified by the robust estimates as point contami-
nations. Therefore, robust portfolios are primarily designed for long run investments. We
note that the notion of “long run” may vary across markets and to account for changes in
the economy, some periodic rebalancing of the portfolio may still be needed. It is expected
from a robust investment to yield higher gains in the long run and to require a smaller
number of updates.
There is no universally accepted best strategy for portfolio management. Best strategy
will change with investor risk aversion, portfolio target return or standard deviation.
Among many others, we consider the popular strategy followed by institutional investors
that monitors a portfolios at an annual (or monthly) frequency and then rebalances only if
the allocation to an asset shifts more than some threshold (5%, 1%). We do not consider
additional factors when implementing the rebalancing strategies, such as trading costs or
cost of time spent which would reduce the return of the portfolio. However we keep track
of the rebalancing frequency of each manager and are able to draw some conclusions based
on their number of rebalancings.
Summarizing, in this paper we address both problems of composing and managing
portfolios, given a set of financial instruments. We do not address the issue of choosing
the component assets. We robustly estimate the data multivariate distribution using pair-
copulas obtaining the inputs which will define the robust efficient frontier. The trajectory
of a target return and the minimum risk portfolios will be managed by twelve managers
during a 2-years period of out-of-sample investigation. We use data from emerging mar-
kets because of the higher volatility of these stock markets and their greater potential
for interdependence with the major markets. More specifically, we use six-dimensional
contemporaneous daily log-returns from the most traded Brazilian stocks, due to Brazil’s
important position among emerging equity markets. The robust portfolios are compared
to their classical version based on the sample empirical estimates,
The contributions of this paper are three fold: (i) we introduce and investigate the per-
formance of pair-copulas based robust portfolios; (ii) we investigate 12 managing strategies
aiming to keep (or restore) the portfolio target, to guarantee the same risk aversion level;
(iii) we illustrate the ideas using Brazilian data.
Findings in the paper are striking and convincing. We found that despite the invest-
ment type, the robust methodology always outperform the classical version. We are able
to determine the best rule for restoring the portfolio to its original balance and keep the
allocations optimal. We show that the best strategy depends on the investor risk profile,
3
and that pair-copulas based robust minimum risk portfolios monitored by a manager which
checks its composition twice a year provides the best long run investment. Additional ex-
ercises are not provided because the aim of the paper is to find best portfolio composition
along with best long run managing strategy for a given set of financial instruments.
The remaining of this paper is organized as follows. In Section 2 we briefly consider the
classical mean-variance methodology for obtaining efficient portfolios, and briefly review
the definitions of pair-copulas and robust estimates. In Section 3 we define 12 strategies
for portfolio monitoring. In Section 4 we analyze two 6-dimensional data sets and assess
the performance of classical and robust target and minimum risk portfolios. In Section 5
we discuss and summarize the results. Section 6 provides the references.
2 Statistical Methodologies
Derived from simple mathematical terms relating the expected return and risk of a port-
folio, Markowitz’s optimization procedure (Markowitz, 1952) for obtaining the efficient
frontier can be considered the most widely used result of modern economics. Based on
the idea that one should diversify to reduce risk, for a given expected return, the portfolio
theory minimizes risk. To this end the theory considers not only the means and variances
of a set of securities, but also their covariances. However, the true data generating process
is unknown and the inputs for the mean-variance algorithm must be estimated.
2.1 Classical allocations
Markowitz mean-variance optimization is a quadratic optimization problem, whose classi-
cal inputs are the sample mean and the sample covariance matrix, the maximum likelihood
estimates (MLE) under multivariate normality. We call the portfolios obtained using the
classical inputs as classical portfolios.
One of the most important causes of limitation of the method in practice is the lack of
optimality presented by classical estimates in the financial environment. This problem was
studied by Klein and Bawa (1976), Jobson and Korkie (1980, 1981), Canela and Collazo
(2007), among others. Following Markowitz work, a large part the literature was devoted
to obtain reliable alternatives for the sample mean and the sample covariance matrix.
2.2 Robust pair-copulas based allocations
In the last decade, copulas have been widely used in finance (see for example, Embrechts
et al. (1999); Demarta and McNeil (2005); and Gatzert et al. (2008)). The reason for
this popularity is the inadequacy of the multivariate normal distribution when modeling
4
financial data. Any multivariate distribution may be decomposed on its marginal uni-
variate distributions and a copula, which contains all information about the dependence
structures linking the margins (Nelsen (2007); Joe (1997)). Therefore, modeling the data
through copulas allows one to obtain estimates of any characteristic of the distribution,
including Markowitz inputs, the mean vector and the covariance matrix. We call these
estimates as copulas- (or later pair-copulas-) -based estimates.
Consider a continuous random vector X1, . . . , Xd with joint cumulative distribution
function (c.d.f.) H(x1, . . . , xd) and marginal distributions F1, . . . , Fd. Sklar’s theorem
(Sklar, 1959) ensures that there exists a d-copula C such that for all (x1, . . . , xd) ∈
[−∞,∞]d
H(x1, . . . , xd) = C(F1(x1), . . . , Fd(xd)). (1)
Conversely, if C is a d-copula and F1, . . . , Fd are c.d.f.s, the function H defined by (1) is a
d-dimensional distribution function with margins F1, . . . , Fd. Furthermore, if all marginal
c.d.f.s are continuous, C is unique. A d-dimensional copula C is a d-dimensional c.d.f. on
[0, 1]d with standard uniform marginal distributions.
When C is absolutely continuous, taking partial derivatives of (1) one obtains
h(x1, · · · , xd) = c(F1(x1), · · · , Fd(xd))
d∏
i=1
fi(xi) (2)
for some d-dimensional copula density c. This expression is the basis for the usual two
steps inference approach, where firstly the marginal estimates are obtained, and then the
copula parameters are estimated (Joe and Xu (1996), Joe (1997)).
The equation (2) allows for tailored marginal modeling considering all characteristics
of each Fi, including the mean, standard deviation, skewness, kurtosis and any type of
short and long memory serial dependence, plus a search for the best fit for the dependence
structure through a large number of copula families that may be considered. This results
in flexible multivariate distributions well fitted to the data.
A copula function is invariant under monotone increasing transformations of X, making
copula-based dependence measures interesting scale-free tools for studying dependence.
An important copula-based dependence concept is the coefficient of upper tail dependence,
λU , defined as
λU = limα→0+
λU(α) = limα→0+
Pr{X1 > F−1
1(1 − α)|X2 > F−1
2(1 − α)} ,
provided a limit λU ∈ [0, 1] exists. If λU ∈ (0, 1], then X1 and X2 are said to be asymp-
totically dependent in the upper tail. If λU = 0, they are asymptotically independent.
5
Similarly, the lower tail dependence coefficient, λL, is given by
λL = limα→0+
λL(α) = limα→0+
Pr{X1 < F−1
1(α)|X2 < F−1
2(α)} ,
provided a limit λL ∈ [0, 1] exists. It follows that
λU = limu↑1
C(u, u)
1 − u, where C(u1, u2) = Pr{U1 > u1, U2 > u2} and λL = lim
u↓0
C(u, u)
u.
The coefficient of tail dependence measures the amount of dependence in the upper
(lower) quadrant tail of a bivariate distribution. In finance, it is related to the strength of
association during extreme events. The copula derived from the multivariate normal dis-
tribution does not have tail dependence. Therefore, if this copula is assumed for modeling
log-returns, for many pairs of variables it will underestimate joint risks.
More flexibility may be gainned by considering pair-copulas models. Pair-copulas is
an hierarquichal decomposition of a d-copula into a cascade of of potentially different
bivariate copulas. It was originally proposed by Joe (1997), and later discussed in detail
by Bedford and Cooke (2001, 2002), Kurowicka and Cooke (2006) and Aas et al. (2007).
The composing bivariate copulas may vary freely, with respect to choice of the parametric
families and parameter values. Therefore, all types and strengths of dependence can be
covered.
For large d, the number of possible pair-copula constructions is very large. Bedford
and Cooke (2001) introduced a systematic way to obtain the decompositions, the so called
regular vines. These graphical models help understanding the conditional specifications
made for the joint distribution. Two special cases are the canonical vines (C-vines) and
the D-vines. Again, the success of estimation procedure starts with good marginal fits (see
Frahm, Junker, and Schmidt, 2004), which typically pose no difficulties. MLE estimates
may be computed at both steps.
However, occasional atypical points may occur in finance, and they may corrupt the
classical estimates of the dependence structure. Mendes and Accioly (2011) proposed to
robustly estimate pair-copula models using the Weighted Minimum Distance (WMDE)
and the Weighted Maximum Likelihood (WMLE) estimates. They are based on either a
redescending weight function or on a hard rejection rule. Robust estimation of a pair-
copula model is also performed at the level of bivariate data, and details about the robust
estimates found in Mendes et al. (2007).
The WMDE minimize some selected weighted distance from the empirical copula, a
goodness of fit statistics. The WMLE result from a two-step procedure. In the first
step, outlying data points are identified by a robust covariance estimator and receive
zero weights, and in the second step the copula MLE are computed for the reduced data.
6
There are many high breakdown point estimators of multivariate location and scatter that
could be used in this preliminary phase. We use the robust Stahel-Donoho (SD) estimator
based on projections (Stahel, 1981 and Donoho, 1982) which is implemented in the free
R software. This affine equivariant estimator possesses high breakdown point, a bounded
influence function, all good robustness properties which are expected to carry over the
weights of the portfolios.
3 Portfolio Managing (Rebalancing: when and why)
Rebalancing an efficient portfolio means to update the data and re-evaluate the efficient
frontier obtaining a new set of weights defining the assets’ new allocations according to
the preferred risk exposure. Although involving costs, the task is performed because one
wants to bring the assets’ allocations back to the investor’s level of risk tolerance and
expected return.
The frequency of rebalancing is usually pre-determined and follows some rule. Many
rebalancing strategies may be formulated. We define 12 rebalancing strategies through
a list of explicit rules to follow, aiming to represent the various degrees of managers’
risk tolerance, some expressing a very rational line of thinking, others representing some
subjective approach. They determine how frequently the portfolio is monitored and also
which risk measure is being controlled. We do not address two issues. One, if a portfolio
should be rebalanced to its target or to a new allocation suggested by the economy. Two,
we do not consider rebalancing costs (transaction costs, taxes, time, labor, and so on),
which are difficult to quantify, but we do keep track of the number of rebalancing events.
The rules for rebalancing are:
Manager 1: Does nothing. Keeps the portfolio’s allocations unchanged.
Manager 2: Rebalances the portfolio weekly (5 business days).
Manager 3: Rebalances the portfolio monthly (21 business days).
Manager 4: Rebalances the portfolio every three months (63 business days).
Manager 5: Rebalances the portfolio every semester (126 business days).
Manager 6: Follows the Drawdown. If the portfolio’s Drawdown duration reaches three
days, he/she rebalances.
Manager 7: Follows the Drawdown. If the portfolio’s Drawdown duration reaches six
days, he/she rebalances.
7
Manager 8: Follows the Drawdown. If the portfolio’s Drawdown value is equal to or less
than the 0.25 quantile of the portfolio’s Drawdown empirical distribution, he/she
rebalances.
Manager 9: Follows the Value-at-Risk (VaR). He/she inspects the portfolio monthly (each
21 days) and rebalances if there were at least three returns smaller that the current
VaR0.01 in the last 100 days, or if the current VaR0.01 was not violated during the
last 300 days.
Manager 10: Re-evaluates the allocations monthly (each 21 days) and rebalances if for
any asset its updated allocation differs from its current allocation by 5% or more.
Manager 11: Re-evaluates the allocations monthly (each 21 days) and rebalances if for
any asset its updated allocation differs from its current allocation by 10% or more.
Manager 12: Re-evaluates the allocations every semester (each 126 days) and rebalances
if for any asset its updated allocation differs from its current allocation by 10% or
more.
In the analysis of Section 4 we investigate the performance of the managers when
monitoring a minimum risk and a fixed target portfolio. The goal is to find the best
rebalancing strategy which would provide the higher accumulated gain along a testing
period, while maintaining the main characteristic of each investment, namely, to possess
minimum risk, and to attain some target return, respectively. We assume that the assets
chosen to form the portfolio will be available during the validation period, and that they
will not be exchanged.
At each portfolio update, managers 6, 7, 8, and 9 also update the corresponding risk
measure. The VaR0.01 is computed as the 0.01-quantile of the empirical distribution of
the portfolio. The Drawdown is computed as the sum of losses occurring in a sequence
of negative daily returns. In other words, the drawdown is defined as the percentual
accumulated loss due to a sequence of drops in the price of an investment (Grossman
and Zhou (1993); Chekhlov et al. (2003)). It is a flexible risk measure collected over
non-fixed time intervals and provides a different perception of the risk and price flow of
this investment. Thus, for managers 6, 7 and 8 there is no periodicity in his/her behavior.
Rule followed by Manager 9 was somehow suggested by the data analyzed in Section 4,
which presented a period of high turbulence at the end of the estimation period, yielding
more extreme risk measures. So the rule tried to detect if the economy has changed either
back to a less volatile period, or even increasing to a more volatile one. We note that the
type I error probability in both cases are approximately 6% and 5%.
8
Managers 10, 11, and 12 follow rules which are popular among private investors and
also managers in financial institutions.
4 Empirical Analyzes
As stated in the Introduction, our objective in this empirical investigation is two-fold. For
a given data set, firstly we want to compare the performance of the robust and classical
portfolios (as defined in Section 2). Secondly, we want to assess the performance of the
twelve managers, finding the best rule for each portfolio type.
The daily returns were computed from 10 years of transaction prices on the six most
traded Brazilian stocks “Vale do Rio Doce”, “Petrobras”, “Usiminas”, “Banco do Brasil”,
“Eletrobras” and “Tim” provided by BOVESPA. We split the data in two periods, one
for estimation (in-sample 8-years period from June/22/2001 through June/19/2009) and
the 2-years out-of-sample period from June/20/2009 through June/24/2011.
We select two portfolio types for the investigations: the Minimum Risk, and a Fixed
Target. Since the choice of the target value is subjective, we decided to select a portfolio lo-
cated approximately at one third along the efficient frontier, for which all component assets
typically had a positive weight contribution. Actually, this target portfolio could represent
a popular choice for an investment in emerging markets, due to its low risk when compared
to the Maximum Risk portfolio, and to its interesting daily target return of 0.1040% cor-
responding to an annualized log-return of almost 30%. We note that the in-sample average
daily return for the six stocks are respectively (0.1164, 0.0935, 0.1600, 0.1159, 0.0442,−0.0097).
The portfolios computed at the end of the estimation period will be called baseline
portfolios. To obtain the classical portfolios’ allocations we compute the mean-variance
inputs using the classical sample estimates. To obtain the allocations of the robust port-
folios we carry on the two-steps estimation method, as follows.
Initially the unconditional distributions of each series is estimated. We fit by maximum
likelihood a skew-t distribution (Hansen, 1994) to the six 8-years series of log-returns, and
from the estimated c.d.f.s we obtain the pseudo uniform(0, 1) data. Estimates are not
shown here, but available to the reader by request. The marginal fits were carefully
checked since a poor fit would result in the probability integral transformed values not
being standard uniform or i.i.d.. As a consequence, any copula model would be mis-
specified.
9
V & PBB7(1.48, 0.94)
(0.477, 0.404)
P & UBB7(1.32, 0.81)
(0.424, 0.312)
AAAU
��
��
AAAU
U & BBBB7(1.31, 0.74)
(0.393, 0.302)
��
��
AAAU
BB & EN (0.496)
(0, 0)
��
��
AAAU
E & Tt(0.48, 24)
(0.007, 0.007)
��
��
V&U|P
t(0.282, 7)
(0.067, 0.067)
AAAU
P&BB|U
t(0.223, 9)
(0.030, 0.030)
AAAU
��
��
U&E|BB
BB1(0.271, 1.132)
(0.104, 0.155)
AAAU
��
��
BB&T|E
t(0.287, 10)
(0.031, 0.031)
��
��
V&BB|U,P
N(0.088)
(0, 0)
AAAU
P&E|BB,U
F(1.012)
(0, 0)
AAAU
��
��
U&T|BB,E
N(0.210)
(0, 0)
��
��
V&E|BB,U,P
t(−0.0223, 21)
(0.0001, 0.0001)
P&T|BB,U,E
t(0.102, 11)
(0.009, 0.009)
AAAU
��
��
V&T|BB,U,E,P
F(0.4661)
(0, 0)
Exhibit 1: D-vine decomposition. Best copula fits and their parameters estimates. The third row
inside the boxes gives the tail dependence coefficients (λL, λU). Notation in figure: Copulas: N:
Normal; t: the t-copula; F: Frank. Data: V: Vale; P: Petrobras; U: Usiminas; BB: Banco do
Brasil; E: Eletrobras; T: Tim.
Five unconditional and ten conditional bivariate copulas compose the D-vine. The
parametric copula families considered included the Normal, t-student, BB7, BB1, Clayton,
Gumbel, Tawn, Frank, and the product copula. These copula families cover all possible
combinations of values for the lower and upper tail dependence coefficients. In addition,
asymmetric dependence may be modeled by the Tawn copula, a non-exchangeable de-
pendence structure. To find the best copula fit, we compared the value of the penalized
log-likelihood (AIC), examined the pp-plots based on the estimated and the empirical
10
copula, and computed a GOF test statistic (Genest and Remillard (2005) and Genest et
al. (2007), Berg (2008)).
The D-vine decomposition, along with the robust copula fits with parameter estimates
and tail dependence coefficients (λL and λU), are shown in Exhibit 1. Pairs in the first level
are those possessing higher correlations. The robust fits reflect the pattern of the majority
of days and are expected to provide better inputs for the mean-variance algorithm and
therefore more consistent long-run portfolios.
Likewise in Mendes et al. (2010), we compute the rank correlations provided by the
pair-copula decomposition. The pair-copula-based robust estimates along with the skew-
t location and scale estimates provide the inputs for the mean-variance algorithm. We
run the long-only MV optimization algorithm and construct the classical and the robust
pair-copula-based efficient frontiers.
For evaluating the performance of the classical and the robust pair-copula based port-
folios we now assume that the four portfolios are managed by the twelve managers along
the 2-years validation period. The rules are followed and portfolios are updated according
to them.
Firstly, we fix the portfolio type (statisticalmethodology and investment objective) and
find the best rebalancing strategy for this financial instrument. To this end we (a) count
the percentage of time its accumulated gain under some rebalancing approach is higher
or equal than the other competitors; (b) test if the mean difference between accumulated
gains from every pair of managers is statistically different from zero, using a one-sided
robust nonparametric t-test (Wilcoxon rank sum test) at the 1% level.
We found that for Classical Target Portfolio, blindly rebalancing each 5 days (Manager
2) leads to consistently significantly lower accumulated gains. Keeping the allocations
unchanged for two years (Manager 1) is the second worst thing one can do (wins over only
Manager 2). For the Classical Target Portfolio the best strategy comes from Manager
10, which inspects the allocations each 21 days (every month) and rebalances only if any
allocation has changed by 5%. The proportions of times Manager 10 portfolio accumulated
gains is higher than the others is typically around 80%, and highly statistically significant,
see Figure 1. Figure 1 shows the differences between the accumulated gain from Manager
10 and all other managers. For this data set the resulting portfolio coincides with the one
from Manager 3 (which blindly rebalance each month).
For the Robust Target Portfolio the worst thing one can do is to blindly frequently
rebalance the portfolio each 5 days (Manager 2). This is in line with its long run investment
characteristic implied by the robust methodology. However, since the economy changes
and there is a target to follow, some type of re-allocations are needed, and the best manager
11
0 100 200 300 400 500
-2-1
01
2
M 10 - M 1
0 100 200 300 400 500
-2-1
01
2
M 10 - M 2
0 100 200 300 400 500
-0.4
-0.2
0.0
0.2
0.4
M 10 - M 3
0 100 200 300 400 500
-10
1
M 10 - M 4
0 100 200 300 400 500
-2-1
01
2
M 10 - M 5
0 100 200 300 400 500
-10
1
M 10 - M 6
0 100 200 300 400 500
-2-1
01
2
M 10 - M 7
0 100 200 300 400 500
-2-1
01
2
M 10 - M 8
0 100 200 300 400 500
-10
1
M 10 - M 9
0 100 200 300 400 500
-1.0
-0.5
0.0
0.5
1.0
M 10 - M 11
0 100 200 300 400 500
-2-1
01
2 M 10 - M 12
Classical Target Portfolio -- The difference: Manager 10 - All Managers
Figure 1: For the Classical Target portfolio figures show the differences between the accu-
mulated gains under Manager 10 and all other managers.
for the Robust Target Portfolio turned out to be Manager 9, which strategy is based on the
VaR value. Even though he/she inspects monthly, only 14 investment updates were done
(in contrast, for the Classical Target portfolio, the winner Manager 10 has rebalanced 23
times). Figure 2 shows the differences between the accumulated gain from Manager 9 and
all other managers for this portfolio.
On the other hand, when it comes to Classical Minimum Risk Portfolios, keeping
the allocations unchanged for two years (Manager 1) apparently came out as the best
rebalancing strategy. It lead to significant consistently higher accumulated gains when
compared to the gains from the remaining strategies. However, this result should be
looked at with care since the baseline allocations might not result in an efficient minimum
risk anymore. Figure 3 shows the updated efficient frontier after the two-years validation
period, and the position of the baseline classical minimum risk portfolio in the risk ×
return space.
Thus rebalancing is needed to keep the investment characteristic during its life time.
Manager 11 (which monitors the portfolio monthly with a 10% threshold) provided the
best maintenance strategy, see Figure 4. The second best manager is number 9 which uses
the VaR measure. From Figure 4 it is clear the superiority of these two strategies.
12
0 100 200 300 400 500
-2-1
01
2
M 9 - M 1
0 100 200 300 400 500
-2-1
01
2
M 9 - M 2
0 100 200 300 400 500
-1.5
-0.5
0.5
1.5
M 9 - M 3
0 100 200 300 400 500
-10
1
M 9 - M 4
0 100 200 300 400 500
-2-1
01
2
M 9 - M 5
0 100 200 300 400 500
-10
1
M 9 - M 6
0 100 200 300 400 500
-10
1
M 9 - M 7
0 100 200 300 400 500
-2-1
01
2
M 9 - M 8
0 100 200 300 400 500
-1.5
-0.5
0.5
1.5
M 9 - M 10
0 100 200 300 400 500
-1.5
-0.5
0.5
1.5
M 9 - M 11
0 100 200 300 400 500
-2-1
01
2
M 9 - M 12
Robust Target Portfolio -- The difference: Manager 9 - All Managers
Figure 2: For the Robust Target portfolio figures show the differences between the accumu-
lated gains under Manager 9 and all other managers.
Best strategy for rebalancing the Robust Minimum Risk portfolios came from Manager
12, whose strategy is to re-evaluate the allocations each 6 months and rebalance whenever
allocations change by 10% (see Figure 5). This is also in line with the long run characteris-
tic of robust procedures. We note that Manager 12 just rebalanced once and outperformed
Manager 1 which kept the allocations unchanged for two years. Interestingly, the second
best manager is Manager 9, which controls the Value-at-Risk of the portfolio.
Secondly, we compare the performances of the same type of classical and robust in-
vestments driven by their best managers. The upper plot in Figure 6 shows the differences
between the accumulated gains from the Robust Target & Manager 9 portfolio and the
Classical Target & Manager 10 portfolio. We observe that even though the target is the
same the robustly estimated pair-copulas based portfolios outperform the classical ver-
sion. The classical version was updated 23 times whereas the robust one only 14 times.
The lower plot in Figure 6 shows the outstanding superiority of the Robust Minimum
Risk investment & Manager 12 (rebalanced once) over the classical version & Manager 11
(rebalanced 9 times).
Following a suggestion from a referee, we now investigate the performance of the ro-
bustly estimated pair-copulas method combined with the proposed managers for different
13
Risk (standard deviation)
Exp
ecte
d R
etur
n
0.020 0.022 0.024 0.026 0.028
0.00
080.
0009
0.00
100.
0011
Baseline Classical Minimum Risk Portfolio
Figure 3: Updated efficient frontier and baseline Classical Minimum Risk portfolio.
financial instruments in the same country. Two stocks (Eletrobras and Tim) were replaced
by the series IMA-C and IMA-S, respectively a long-term inflation-indexed Brazilian trea-
sury bonds index and a floating rate Brazilian Treasury bill index, both computed by
the Brazilian Association of Financial Institutions, ANBIMA (www.anbima.com.br). The
IMA-S consists of the price changes of Letras Financeiras do Tesouro (LFT), which are
zero-coupon shorter term securities whose interest rate is compounded daily using the
average treasury repo market rates computed by the Brazilian Central Bank (the SELIC
rate) and that is paid according to this accruing in one single payment at the maturity
date.
We again split the data in two parts, a in-sample 8-years period for data estimation
and the 2-years out-of-sample period for models validation. The Fixed Target portfolio
now has a daily target return of 0.17%. The new series present positive in-sample average
daily return, respectively, 0.0786 and 0.0617. Their main characteristic is their much
smaller sample standard deviations, respectively, 0.204 and 0.054, when compared to
those computed for the remaining four stocks, respectively, 2.542, 2.452, 3.185 and 3.033.
The same estimation steps were followed and optimal portfolios’ weights computed. The
difference now between the classical and the copula based efficient frontiers is not so
dramatic, but the robust-pc based curve is still located above and at the left of the
14
0 100 200 300 400 500
-0.5
0.0
0.5
M 11 - M 1
0 100 200 300 400 500
-0.6
-0.2
0.2
0.6
M 11 - M 2
0 100 200 300 400 500
-0.6
-0.2
0.2
0.6
M 11 - M 3
0 100 200 300 400 500
-0.6
-0.2
0.2
0.6
M 11 - M 4
0 100 200 300 400 500
-0.6
-0.2
0.2
0.6
M 11 - M 5
0 100 200 300 400 500
-0.6
-0.2
0.2
0.6
M 11 - M 6
0 100 200 300 400 500
-0.6
-0.2
0.2
0.6
M 11 - M 7
0 100 200 300 400 500
-0.6
-0.2
0.2
0.6
M 11 - M 8
0 100 200 300 400 500
-0.6
-0.2
0.2
0.6
M 11 - M 9
0 100 200 300 400 500
-0.6
-0.2
0.2
0.6
M 11 - M 10
0 100 200 300 400 500
-0.6
-0.2
0.2
0.6
M 11 - M 12
Classical Minimum Risk Portfolio -- The difference: Manager 11 - All Managers
Figure 4: For the Classical Minimum risk portfolio figures show the differences between
the accumulated gains under Manager 11 and all other managers.
classical version. In particular, the weights defining the Minimum Risk portfolios are very
close, both methods allocating around 95% on the IMA-S, due to the obvious reason.
The target portfolios had almost all weight allocated to three components, the Vale and
Usiminas stocks and the IMA-C. We repeated the same strategies for evaluating and
comparing the performance of classical and pc-based portfolios.
We found that the best rebalancing strategy for the Classical Target Portfolio came
from Manager 6 which updates the optimal weights whenever three consecutive losses are
observed. The proportions of times Manager 6 Classical Target Portfolio accumulated
gains is higher than all others is around 84%, highly statistically significant.
For the Robust Target Portfolio the best manager turned out to be Manager 7, which
strategy is to update whenever there is a sequence of 6 negative returns. Only one update
was made, whereas for the Classical Target portfolio, the winner Manager 6 has rebalanced
19 times. For the Robust Target Portfolio he second best performance was provided by
Manager 4 which inspected at the end of 63 business days, having carried out seven
investment updates.
On the other hand, the Classical Minimum Risk Portfolio driven by Manager 12 pro-
vided higher accumulated gains than all others during time percentages varying between
15
0 100 200 300 400 500
-1.5
-0.5
0.5
1.5
M 12 - M 1
0 100 200 300 400 500
-10
1
M 12 - M 2
0 100 200 300 400 500
-10
1
M 12 - M 3
0 100 200 300 400 500
-10
1
M 12 - M 4
0 100 200 300 400 500
-10
1
M 12 - M 5
0 100 200 300 400 500
-10
1
M 12 - M 6
0 100 200 300 400 500
-10
1
M 12 - M 7
0 100 200 300 400 500
-10
1
M 12 - M 8
0 100 200 300 400 500
-10
1
M 12 - M 9
0 100 200 300 400 500
-10
1
M 12 - M 10
0 100 200 300 400 500
-10
1
M 12 - M 11
Robust Minimum Risk Portfolio -- The difference: Manager 12 - All Managers
Figure 5: For the Robust Minimum Risk portfolio figures show the differences between the
accumulated gains under Manager 12 and all other managers.
67% and 97%. This manager has inspected the allocations every semester using a fluctu-
ation margin of 10%, but has performed only one update. For the minimum risk portfolio
we observed that the weights distribution along the validation period was very stable,
explaining the second place been occupied by three managers which performed zero up-
dates, namely managers 1, 7, and 8. We note that the minimum risk portfolio has allocated
approximately 95% weight to the low risk IMA-S.
Best strategy for rebalancing the Robust Minimum Risk Portfolio came from Manager
10 , which re-evaluates the allocations monthly and rebalances if the updated allocation
changes by 5% or more for any asset. The number of updates carried out was 9.
Finally, we compare the performances of the same type of classical and robust invest-
ments driven by their best managers. Likewise Figure 6, Figure 7 shows the differences
between the accumulated gains from the Robust Target & Manager 7 portfolio and the
Classical Target & Manager 6 portfolio (upper row). In the lower row, Figure 7 shows
the differences between the accumulated gains from Robust Minimum Risk investment &
Manager 10 and the classical version & Manager 12.
16
0 100 200 300 400 500
-2-1
01
2
Difference between the accumulated gains from the robust and classical target portfolios
0 100 200 300 400 500
-20
2
Difference between the accumulated gains from the robust and classical minimum risk portfolios
Figure 6: Difference between the accumulated gains from the best robust and best classical
portfolios.
5 Discussions
As usual, the conclusions drawn in this paper only apply to data possessing similar char-
acteristics. However, the results and discussions may shed some light on the largely
discussed topic of portfolio allocation and rebalancing strategies, and may be easily tested
and extended to other investments based on different asset characteristics (expected re-
turn, volatility and correlations). More importantly, this work has shown that the robustly
estimated pair-copulas based mean-variance inputs are more accurate thus increasing the
chances of producing a financial instrument which will truthfully yield what was expected
from it.
From the analyzes carried on it was clear the superiority of the robust portfolios. But
even a robust portfolio must be properly managed. According to the empirical analysis
of the first data set composed only by daily stock returns, for the Robust Minimum Risk
portfolio the second best manager with respect to higher accumulated gains is Manager
9 (which uses the VaR). However, Manager 9 has done 9 rebalancings, whereas Manager
12 carried on only one after one year and a half. One may be tempted to conclude that
the good performance of the winner may be just due to the changes in the economy which
17
0 100 200 300 400 500
-2-1
01
2
Difference between the accumulated gains from the robust and classical target portfolios
0 100 200 300 400 500
-0.1
5-0
.05
0.05
0.15
Difference between the accumulated gains from the robust and classical minimum risk portfolios
Figure 7: For the new data set, the difference between the accumulated gains from the best
robust and best classical portfolios.
may have taken the portfolio away from the efficient frontier. It was not the case. Figure
8 illustrates and shows the positions of the robust minimum risk portfolios from both
managers (based on weights from last rebalancing and using the entire validation period
data) along with the updated efficient frontier. We observe that the portfolios are very
close and still close to the curve, with the point risk×return from Manager 9 (triangle in
pink) even a little bit higher. The success of this procedure may be credited to the timing
of Manager 12 combined with the good stability of the robust pair-copulas method of
portfolio construction. Actually, for the second data set used based on less volatile assets,
Manager 12 was also the best option for the Classical Minimum Risk portfolio.
Manager 9 was also the second best option for the Classical Minimum Risk portfolio,
and the best one for the Robust Target. Thus, one may say that controlling the Value-at-
Risk is also an efficient strategy. Another issue is whether or not the target portfolio should
be restored to the same target. We did not address this problem, but this consideration will
certainly not change the result towards the excellent performance of the robust method.
Many other managers’ rules could be defined. Another strategy that seems promising
is a variation of the rules followed by managers 10, or 11, or 12, where the threshold
determining the need for rebalancing would not be fixed for all assets. Instead, it would
18
Risk (standard deviation)
Exp
ecte
d R
etur
n
0.020 0.022 0.024 0.026 0.028 0.030
0.00
080.
0009
0.00
100.
0011
Updated Robust Minimum Risk portfolios
(Blue: Manager 12; Pink: Manager 9)
Figure 8: Updated efficient frontier and updated robust minimum risk portfolios from
Manager 12 and Manager 9.
vary according to each asset weight or importance in the composition. In addition, it
deserves further investigation the actual value of the threshold. Other values beyond the
assumed 5% and 10% may lead to better performances.
Another important result drawn from this empirical analysis is concerned with costs.
We found that despite portfolio type, robust portfolios typically demand a smaller number
of updates lowering costs.
Finally, we found that despite the rebalancing rule, the robust portfolios always out-
perform the classical versions. Figure 9 shows the differences between the accumulated
gains from the robust and the calssical Target portfolios, having fixed the managing rule.
Given the same manager and the same portfolio target, the robust method is always su-
perior to the classical method. This is also true for the Minimum Risk portfolios, and the
outstanding performance of the Robust Minimum Risk portfolio for all fixed managers
can be seen in Figure 10.
[Figure 10 around here]
In summary, observing that the dependence between assets go beyond the linear cor-
19
0 100 200 300 400 500
-3-2
-10
12
3
M 1
0 100 200 300 400 500
-3-2
-10
12
3
M 2
0 100 200 300 400 500
-3-2
-10
12
3
M 3
0 100 200 300 400 500
-3-2
-10
12
3
M 4
0 100 200 300 400 500
-3-2
-10
12
3
M 5
0 100 200 300 400 500
-3-2
-10
12
3
M 6
0 100 200 300 400 500
-3-2
-10
12
3
M 7
0 100 200 300 400 500
-3-2
-10
12
3
M 8
0 100 200 300 400 500
-3-2
-10
12
3
M 9
0 100 200 300 400 500
-3-2
-10
12
3
M 10
0 100 200 300 400 500
-3-2
-10
12
3
M 11
0 100 200 300 400 500
-3-2
-10
12
3
M 12
Target Portfolios: difference between Robust and Classical methods and for all fixed managers
Figure 9: For the Target portfolio figure shows the differences between the robust and the
classical accumulated gains keeping fixed the manager.
Figure 10: For the Minimum Risk portfolio figure shows the differences between the robust
and the classical accumulated gains keeping fixed the manager.
relation, in this paper we proposed modeling log-returns data using robustly estimated
pair-copula models. The method is appealing simple and able to handle contaminations
that may occur when working with financial data. We illustrated the idea in the context of
emerging stock markets using two data sets composed by the most liquid Brazilian stocks,
a long-term inflation-indexed Brazilian treasury bonds index and a floating rate Brazilian
Treasury bill index. The empirical analyzes carried on in this paper indicated that for any
type of portfolio we are able to find the best manager strategy. Moreover, they indicated
that the robustly estimated pair-copulas based portfolios always outperform the classical
versions despite the managing strategy.
Methodology seems promising for any risk profile investor. The interested reader (or
investor) may easily tailor these ideas to his/her needs, repeating these exercises using
other data sets and considering other investor risk aversion levels, and even defining new
rules for managing. We are very confident that it will always be a combination of manager
and a robust portfolio outperforming its classical version. The authors will be happy to
20
compute the robust estimates and check the performance of any portfolio for any data set
the reader shall have. Future research may include the investigation of different managers’
rules for other types of portfolios.
Indeed, we have a final recommendation. If the objective is a minimum risk long
run investment the best one can do is to allocate the assets using robustly estimated
pair-copulas estimates and inspect the portfolio each 6 months, rebalancing whenever
allocations change by 10%. This will guarantee the investment characteristics and provide
the cheapest managing strategy.
Acknowledgements. The first author thanks the financial support from CNPq. Both
authors gratefully acknowledge COPPEAD research funds.
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